{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Random Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Superposition of Random Signals\n",
    "\n",
    "The superposition of two random signals \n",
    "\n",
    "\\begin{equation}\n",
    "y[k] = x[k] + n[k]\n",
    "\\end{equation}\n",
    "\n",
    "is a frequently applied operation in statistical signal processing. For instance, to model a measurement procedure or communication channel as superposition of the desired signal with noise. We assume that the statistical properties of the real-valued signals $x[k]$ and $n[k]$ are known. We are interested in the statistical properties of $y[k]$, as well as the joint statistical properties between the signals and their superposition $y[k]$. It is assumed for the following that $x[k]$ and $n[k]$ are drawn from wide-sense stationary (WSS) real-valued random processes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cumulative Distribution and Probability Density Function\n",
    "\n",
    "The cumulative distribution function (CDF) $P_y(\\theta)$ of $y[k]$ is given by rewriting it in terms of the joint probability density function (PDF) $p_{xn}(\\theta_x, \\theta_n)$\n",
    "\n",
    "\\begin{equation}\n",
    "P_y(\\theta) = \\Pr \\{ y[k] \\leq \\theta \\} = \\Pr \\{ (x[k] + n[k]) \\leq \\theta \\} =\n",
    "\\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{-\\infty}^{\\theta - \\theta_n} p_{xn}(\\theta_x, \\theta_n) \\; \\mathrm{d}\\theta_x\\,\\mathrm{d}\\theta_n\n",
    "\\end{equation}\n",
    "\n",
    "Its PDF is computed by introducing above result into the [definition](distributions.ipynb#Univariate-Probability-Density-Function) of the PDF\n",
    "\n",
    "\\begin{equation}\n",
    "p_y(\\theta) = \\frac{\\mathrm{d} P_y(\\theta)}{\\mathrm{d}\\theta} = \\int\\limits_{-\\infty}^{\\infty} p_{xn}(\\theta - \\theta_n, \\theta_n) \\; \\mathrm{d}\\theta_n\n",
    "\\end{equation}\n",
    "\n",
    "since the inner integral on the right hand side of $P_y(\\theta)$ can be interpreted as the inverse operation to the derivation with respect to $\\theta$.\n",
    "\n",
    "An important special case is that $x[k]$ and $n[k]$ are uncorrelated. Under this assumption, the joint PDF $p_{xn}(\\theta_x, \\theta_n)$ can be written as $p_{xn}(\\theta_x, \\theta_n) = p_x(\\theta_x) \\cdot p_n(\\theta_n)$. Introducing this into above result yields\n",
    "\n",
    "\\begin{align}\n",
    "p_y(\\theta) &= \\int\\limits_{-\\infty}^{\\infty} p_x(\\theta - \\theta_n) \\cdot p_n(\\theta_n) \\; \\mathrm{d}\\theta_n \\\\\n",
    "&= p_x(\\theta) * p_n(\\theta)\n",
    "\\end{align}\n",
    "\n",
    "Hence, the PDF of the superposition of two uncorrelated signals is given by the convolution of the PDFs of both signals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - PDF of a superposition of two uncorrelated signals\n",
    "\n",
    "The following example estimates the PDF of a superposition of two uncorrelated signals drawn from random processes generating samples according to the [uniformly distributed](important_distributions.ipynb#Uniform-Distribution) white noise model with $a=0$ and $b=1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "K = 100000  # length of random signals\n",
    "\n",
    "# generate random signals\n",
    "np.random.seed(2)\n",
    "x = np.random.uniform(size=K)\n",
    "n = np.random.uniform(size=K)\n",
    "y = x + n\n",
    "\n",
    "# plot estimated pdf\n",
    "plt.figure(figsize = (10, 6))\n",
    "plt.hist(y, 100, density=True)\n",
    "plt.title('Estimated PDF')\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel(r'$\\hat{p}_y(\\theta)$')\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Check the result of the numerical simulation by calculating the theoretical PDF of $y[k]$\n",
    "* What PDF will results if you sum up an infinite number of independent uniformly distributed random signals?\n",
    "\n",
    "Solution: The PDF of both signals $x[k]$ and $n[k]$ is given as $p_x(\\theta) = p_n(\\theta) = \\text{rect}(\\theta - \\frac{1}{2})$, according to the assumptions stated above. The PDF of $y[k] = x[k] + n[k]$ is consequently given by $p_y(\\theta) = \\text{rect}(\\theta - \\frac{1}{2}) * \\text{rect}(\\theta - \\frac{1}{2})$. The convolution of two rectangular signals results in a triangular signal. Hence\n",
    "\n",
    "\\begin{equation}\n",
    "p_y(\\theta) = \n",
    "\\begin{cases}\n",
    "1 - |\\theta -1| & \\text{for } 0 < \\theta < 2 \\\\\n",
    "0 & \\text{otherwise}\n",
    "\\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "The [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) states that a superposition of independent random variables tends towards a normal distribution for an increasing number of superpositions. Numerical simulation for a high bus finite number of independent uniformly distributed random signals yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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    "N = 1000  # number of independent random signals\n",
    "\n",
    "# generate random signals\n",
    "np.random.seed(2)\n",
    "y = np.sum(np.random.uniform(size=(N, K)), axis=0)\n",
    "\n",
    "# plot estimated pdf\n",
    "plt.figure(figsize = (10, 6))\n",
    "plt.hist(y, 100, density=True)\n",
    "plt.title('Estimated PDF')\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel(r'$\\hat{p}_y(\\theta)$')\n",
    "plt.grid()"
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   "source": [
    "### Linear Mean\n",
    "\n",
    "The linear mean $\\mu_y$ of the superposition is derived by introducing $y[k] = x[k] + n[k]$ into the [definition of the linear mean](ensemble_averages.ipynb#Linear-mean) and exploiting the [linearity of the expectation operator](ensemble_averages.ipynb#Properties) as\n",
    "\n",
    "\\begin{equation}\n",
    "\\mu_y[k] = E \\{ x[k] + n[k] \\} = \\mu_x[k] + \\mu_n[k]\n",
    "\\end{equation}\n",
    "\n",
    "The linear mean of the superposition of two random signals is the superposition of its linear means."
   ]
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   "source": [
    "### Auto-Correlation Function and Power Spectral Density\n",
    "\n",
    "The ACF is computed in the same manner as the linear mean by inserting the superposition into the [definition](correlation_functions.ipynb#Auto-Correlation-Function) of the ACF and rearranging terms\n",
    "\n",
    "\\begin{align}\n",
    "\\varphi_{yy}[\\kappa] &= E\\{ y[k] \\cdot y[k-\\kappa] \\} \\\\\n",
    "&= E\\{ (x[k] + n[k]) \\cdot (x[k-\\kappa] + n[k-\\kappa]) \\} \\\\\n",
    "&= \\varphi_{xx}[\\kappa] + \\varphi_{xn}[\\kappa] + \\varphi_{nx}[\\kappa] + \\varphi_{nn}[\\kappa] \n",
    "\\end{align}\n",
    "\n",
    "The ACF of the superposition of two random signals is given as the superposition of all auto- and cross-correlation functions (CCFs) of the two random signals. The power spectral density (PSD) is derived by performing a discrete-time Fourier transform (DTFT) of the ACF\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{yy}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{xn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{nx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{nn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
    "\\end{equation}\n",
    "\n",
    "This can be simplified further by exploiting the symmetry property of the CCFs $\\varphi_{xn}[\\kappa] = \\varphi_{nx}[-\\kappa]$ and the DTFT for real-valued signals as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{yy}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + 2\\,\\Re \\{ \\Phi_{xn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\} + \\Phi_{nn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
    "\\end{equation}\n",
    "\n",
    "where $\\Re \\{ \\cdot \\}$ denotes the real part of its argument."
   ]
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   "metadata": {},
   "source": [
    "### Cross-Correlation Function and Cross Power Spectral Density\n",
    "\n",
    "The CCF $\\varphi_{ny}[\\kappa]$ between the random signal $n[k]$ and the superposition $y[k]$ is derived again by introducing the superposition into the [definition of the CCF](correlation_functions.ipynb#Cross-Correlation-Function)\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{ny}[\\kappa] = E\\{ n[k] \\cdot (x[k-\\kappa] + n[k-\\kappa]) \\} = \\varphi_{nx}[\\kappa] + \\varphi_{nn}[\\kappa]\n",
    "\\end{equation}\n",
    "\n",
    "It is given as the superposition of the CCF between the two random signals and the ACF of $n[k]$. The cross PSD is derived by applying the DTFT to $\\varphi_{ny}[\\kappa]$\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ny}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{nx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{nn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
    "\\end{equation}\n",
    "\n",
    "The CCF $\\varphi_{xy}[\\kappa]$ and cross PSD $\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ can be derived by exchanging the signals $n[k]$ and $x[k]$\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xy}[\\kappa] = E\\{ x[k] \\cdot (x[k-\\kappa] + n[k-\\kappa]) \\} = \\varphi_{xx}[\\kappa] + \\varphi_{xn}[\\kappa]\n",
    "\\end{equation}\n",
    "\n",
    "and\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{xn}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
    "\\end{equation}"
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    "### Additive White Gaussian Noise\n",
    "\n",
    "In order to model the effect of distortions, it is often assumed that a random signal $x[k]$ is distorted by additive normal distributed white noise $n[k]$ resulting in the observed signal $y[k] = x[k] + n[k]$. It is furthermore assumed that the noise $n[k]$ is uncorrelated to the signal $x[k]$. This model is known as [additive white Gaussian noise](https://en.wikipedia.org/wiki/Additive_white_Gaussian_noise) (AWGN) model. \n",
    "\n",
    "For zero-mean random processes and from the properties of the AWGN model, it follows that $\\varphi_{xn}[\\kappa] = \\varphi_{nx}[\\kappa] = 0$ and $\\varphi_{nn}[\\kappa] = N_0 \\cdot \\delta[\\kappa]$. Introducing this into above results for additive random signals yields the following relations for the AWGN model\n",
    "\n",
    "\\begin{align}\n",
    "\\varphi_{yy}[\\kappa] &= \\varphi_{xx}[\\kappa] + N_0 \\cdot \\delta[\\kappa] \\\\\n",
    "\\varphi_{ny}[\\kappa] &= N_0 \\cdot \\delta[\\kappa] \\\\\n",
    "\\varphi_{xy}[\\kappa] &= \\varphi_{xx}[\\kappa]\n",
    "\\end{align}\n",
    "\n",
    "The PSDs are given as the DTFTs of the ACF/CCFs. The AWGN model is frequently assumed in communications as well as in the measurement of physical quantities to cope for background, sensor and amplifier noise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Estimating the Period of a Noisy Periodic Signal\n",
    "\n",
    "For the following numerical example, the disturbance of a harmonic signal $x[k] = \\cos(\\Omega_0 k)$ by unit variance AWGN $n[k]$ is considered. The observed signal is given as\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] = x[k] + n[k]\n",
    "\\end{equation}\n",
    "\n",
    "The ACF $\\varphi_{yy}[\\kappa]$ of the compound signal, as well as the CCF between noise and signal $\\varphi_{ny}[\\kappa]$ is computed and plotted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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    "N = 1024  # length of compound signals\n",
    "M = 25  # period of cosine signal\n",
    "K = 2*M  # maximum lag for ACF/CCF\n",
    "\n",
    "# generate signals\n",
    "x = np.cos(2*np.pi/M * np.arange(N))\n",
    "np.random.seed(2)\n",
    "n = np.random.normal(size=N)\n",
    "# superposition of signals\n",
    "y = x + n\n",
    "\n",
    "# compute and truncate ACF of superposition\n",
    "acf = 1/N * np.correlate(y, y, mode='full')\n",
    "acf = acf[(len(y)-1)-K:len(y)+K]\n",
    "# compute and truncate CCF of superposition and noise\n",
    "ccf = 1/N * np.correlate(n, y, mode='full')\n",
    "ccf = ccf[(len(y)-1)-K:len(y)+K]\n",
    "\n",
    "\n",
    "# plot results\n",
    "kappa = np.arange(-K, K+1)\n",
    "\n",
    "plt.figure(figsize=(10, 10))\n",
    "\n",
    "plt.subplot(311)\n",
    "plt.stem(y, use_line_collection=True)\n",
    "plt.title('Signal')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$x[k]$')\n",
    "plt.axis([0, K, -3, 3])\n",
    "\n",
    "plt.subplot(312)\n",
    "plt.stem(kappa, acf, use_line_collection=True)\n",
    "plt.title('ACF of superposition')\n",
    "plt.xlabel(r'$\\kappa$')\n",
    "plt.ylabel(r'$\\varphi_{yy}[\\kappa]$')\n",
    "plt.axis([-K, K, -.75, 1.1*np.max(acf)])\n",
    "plt.grid()\n",
    "\n",
    "plt.subplot(313)\n",
    "plt.stem(kappa, ccf, use_line_collection=True)\n",
    "plt.title('CCF between noise and superposition')\n",
    "plt.xlabel(r'$\\kappa$')\n",
    "plt.ylabel(r'$\\varphi_{ny}[\\kappa]$')\n",
    "plt.axis([-K, K, -.2, 1.1])\n",
    "plt.grid()\n",
    "\n",
    "plt.tight_layout()"
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   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Derive the theoretic result for the ACF $\\varphi_{xx}[\\kappa]$\n",
    "* Based on this, can you explain the shape of the ACF?\n",
    "* Estimate the periodicity/frequency of the cosine signal from the ACF\n",
    "* What conclusions can you draw from the CCF between noise and signal?\n",
    "\n",
    "Solution: The ACF of the deterministic signal $x[k] = \\cos(\\Omega_0 k)$ is computed via temporal averaging. Since $x[k]$ is periodic, the ACF $\\varphi_{xx}[\\kappa]$ is also periodic. It is hence sufficient to perform the temporal averaging over one period $N = \\frac{2 \\pi}{\\Omega_0}$ with $N \\in \\mathbb{Z}$\n",
    "\n",
    "\\begin{align}\n",
    "\\varphi_{xx}[\\kappa] &= \\frac{1}{N} \\sum_{k=0}^{N-1} x[k] \\cdot x[k-\\kappa] \\\\\n",
    "&= \\frac{1}{N} \\sum_{k=0}^{N-1} \\cos(\\Omega_0 k) \\cdot \\cos(\\Omega_0 (k-\\kappa))\n",
    "\\end{align}\n",
    "\n",
    "Applying the [product-to-sum identity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities) of the cosine function and rearranging terms yields\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xx}[\\kappa] = \\frac{1}{2} \\cos(\\Omega_0 \\kappa) + \\frac{1}{2N} \\sum_{k=0}^{N-1} \\cos(2 \\Omega_0 k - \\Omega_0 \\kappa)\n",
    "\\end{equation}\n",
    "\n",
    "The remaining sum over the phase shifted cosine $\\cos(2 \\Omega_0 k - \\Omega_0 \\kappa)$ is zero since the summation is carried out over two full periods. The ACF of a cosine signal is then given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xx}[\\kappa] = \\frac{1}{2} \\cos(\\Omega_0 \\kappa)\n",
    "\\end{equation}\n",
    "\n",
    "Introducing this into the relations for the AWGN model, the ACF of the superposition is\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{yy}[\\kappa] = \\varphi_{xx}[\\kappa] + N_0 \\delta[\\kappa]  = \\frac{1}{2} \\cos(\\Omega_0 \\kappa) + N_0 \\delta[\\kappa]\n",
    "\\end{equation}\n",
    "\n",
    "Taking the statistical uncertainty due to a finite number of samples into account, this analytic result coincides well with the ACF of the numerical evaluation shown above.\n",
    "\n",
    "The ACF of a periodic signal is also periodic. The ACF $\\varphi_{yy}[\\kappa]$ is composed from a superposition of the ACF $\\varphi_{xx}[\\kappa]$ of the periodic cosine signal and an aperiodic contribution from the additive noise. Hence, the period of the cosine can be estimated from the period of the ACF.\n",
    "\n",
    "It can be concluded from the CCF that the additive noise is not correlated with the cosine signal."
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    "### Example - Denoising by Repeated Averaging\n",
    "\n",
    "Let's assume that we can have access to multiple observations of a noisy deterministic signal $x[k]$. The $i$-th observation $y_i[k]$ is then given by the following AWGN model\n",
    "\n",
    "\\begin{equation}\n",
    "y_i[k] = x[k] + n_i[k]\n",
    "\\end{equation}\n",
    "\n",
    "where $n_i[k]$ denotes the $i$-th sample function of a zero-mean white noise process with PSD $N_0$. It is assumed that the sample functions of the noise process are mutually independent. An estimate $\\hat{x}[k]$ of the deterministic signal $x[k]$ is given by averaging over $N$ observations\n",
    "\n",
    "\\begin{equation}\n",
    "\\hat{x}[k] = \\frac{1}{N} \\sum_{i=0}^{N-1} (x[k] + n_i[k]) = x[k] + \\underbrace{\\frac{1}{N} \\sum_{i=0}^{N-1} n_i[k]}_{n[k]}\n",
    "\\end{equation}\n",
    "\n",
    "The estimate consists of a superposition of the true signal $x[k]$ and noise $n[k]$.\n",
    "\n",
    "The [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio) (SNR) of the estimate is defined as the power of the true signal $x[k]$ divided by the power of the additive noise. Following this definition, the average SNR is given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathrm{SNR} = 10 \\cdot \\log_{10} \\left( \\frac{P_x}{\\sigma_n^2} \\right) \\quad \\text{ in dB}\n",
    "\\end{equation}\n",
    "\n",
    "where $P_x$ denotes the average power of the signal $x[k]$ and $\\sigma_n^2$ the variance of the noise $n[k]$. The SNR of the estimate $\\hat{x}[k]$ is computed in the following to quantify the gain in SNR achieved by averaging. The average power $P_x$ of the signal is assumed to be known, the power of the noise $\\sigma_x^2$ can be computed from its ACF. Repeated application of above findings for the AWGN model yields\n",
    "\n",
    "\\begin{equation}\n",
    "\\sigma_n^2 = \\frac{1}{N^2} \\, ( N \\cdot N_0 ) = \\frac{N_0}{N}\n",
    "\\end{equation}\n",
    "\n",
    "since the scaling $\\frac{1}{N}$ in above sum results in a $\\frac{1}{N^2}$ scaling of the variance and $N$ uncorrelated instances of $n_i[k]$ with power $N_0$ are summed up. The SNR of the estimate follows then as\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathrm{SNR} = 10 \\cdot \\log_{10} \\left( P_x \\right) - 10 \\cdot \\log_{10} \\left( N_0 \\right) + 10 \\cdot \\log_{10} \\left( N \\right) \\quad \\text{ in dB}\n",
    "\\end{equation}\n",
    "\n",
    "By averaging over $N$ observations, an average gain of $10 \\cdot \\log_{10} \\left( N \\right)$ dB in terms of SNR can be achieved.\n",
    "\n",
    "The following numerical example illustrates the denoising of a cosine signal superimposed by unit variance white Gaussian noise. One particular observation $y_i[k]$, the estimate $\\hat{x}[k]$ by averaging over $N$ observations and the resulting SNR is shown."
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   "source": [
    "K = 1024  # number of samples\n",
    "Nmax = 50  # number of observations\n",
    "\n",
    "# generate signals\n",
    "x = np.cos(2*np.pi/50 * np.arange(N))\n",
    "np.random.seed(2)\n",
    "n = np.random.normal(size=(Nmax, K))\n",
    "# AWGN model\n",
    "y = np.tile(x, (Nmax,1)) + n\n",
    "# repeated averaging up to Nmax\n",
    "xhat = np.zeros_like(y)\n",
    "for i in range(Nmax):\n",
    "    xhat[i, :] = 1/(i+1) * np.sum(y[:i+1, :], axis=0)\n",
    "# compute SNR for all averages\n",
    "Px = np.var(x)\n",
    "Pn = np.var(xhat - x, axis=1)\n",
    "SNR = 10*np.log10(Px/Pn)\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 10))\n",
    "\n",
    "plt.subplot(311)\n",
    "plt.stem(y[0, :100], use_line_collection=True)\n",
    "plt.title('One observation')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$y_0[k]$')\n",
    "plt.ylim([-3, 3])\n",
    "\n",
    "plt.subplot(312)\n",
    "plt.stem(xhat[Nmax-1, :100], use_line_collection=True)\n",
    "plt.title('Average over {:2.0f} observations'.format(Nmax))\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$\\hat{x}[k]$')\n",
    "plt.ylim([-3, 3])\n",
    "\n",
    "plt.subplot(313)\n",
    "plt.plot(range(1, Nmax+1), 10*np.log10(Px * range(1, Nmax+1)), '--', label='theory')\n",
    "plt.plot(range(1, Nmax+1), SNR, label='simulated')\n",
    "plt.title('SNR')\n",
    "plt.xlabel('number of averages $N$')\n",
    "plt.ylabel('SNR in dB')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "\n",
    "plt.tight_layout()"
   ]
  },
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   "metadata": {},
   "source": [
    "**Excercise**\n",
    "\n",
    "* How does the SNR change if the instances $n_i[k]$ of the additive noise are mutually correlated?\n",
    "\n",
    "Solution: The gain in terms of SNR will be lower since the mutually correlated AWGNs do not average out in the estimate $\\hat{x}[k]$. For instance, the averaging will have no effect if the samples of the additive noise are equal for all observed signals $y_i[k] = x[k] + n_0[k]$."
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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